We have a question asking When and how does Vader learn that Luke is his son?, but the Emperor knows too, as he says in The Empire Strikes Back:
We have a new enemy, the young rebel who destroyed the Death Star. I have no doubt this boy is the offspring of Anakin Skywalker.
The answer to the aforementioned question shows that in Disney canon comics, Vader found out that the one who destroyed the Death Star was named "Luke Skywalker" only when he sent out Boba Fett on a personal investigation, which the Emperor did not know about. I am looking for an answer that is in Disney canon and takes these comic book developments into account.
So how did the Emperor learn that Luke is Anakin's son? (And for that matter, that he was the one who destroyed the Death Star)?
Points to consider
There's been a lot of misunderstandings in the comments and such, so I want to reiterate a few points:
- Luke is not the only untrained force-sensitive in the Galaxy!!! There are loads of other people in the Galaxy who are force-sensitive. Just because Vader realized that the Force was strong with the pilot who destroyed the Death Star doesn't mean that it was automatically the spawn of Skywalker. Obi-Wan could have found any of thousands of people with latent Force ability and trained them, giving them his old apprentice's ligthsaber. Heck, Asoka was a powerful Force user who was with the Rebels; Vader had just as much evidence that it was her! So just because a powerful Force user destroyed the Death Star does not automatically point to it being Skywalker's son!
- Palpatine lied that Vader killed a pregnant Padmé, but he did not know that Obi-Wan took her to Polis Massa where Luke and Leia were born.
- I have yet to see any evidence that Palpatine has ever had the ability to sense a Force user from across the Galaxy—and in such a way that Vader, standing right next to said Force user, could not—much less identify the lineage of the Force user.
- Per the linked question, Vader found out after sending Boba Fett on a secret mission to learn Luke's name. There isn't any evidence that Palpatine knew about this secret mission, so he must have found out about it some other way than Vader did.